Rebirth and What the Buddha did when nobody was around

Recently I was asked:
Could you please post your views on rebirth/reincarnation? If nothing is or ever was, then why this whole story about The Buddha being all-knowing and able to trace back his births (the story may have been created by his followers later on)?Who or What exactly reincarnates?

In Jed McKenna's last book he puts it well."When someone starts to quote the Buddha, run away, Big Bullshit coming." Or something like that.

I've thought that on *that* day, when the Buddha Attained, got home and told his wife, she said:
"That's nice dear, now go wash up, we have a dinner party to go to..."

One the one hand, it is easy to dismiss stories related about one who may or may not have existed thousands of years ago; and most stories written about him centuries after his death.

Nisargadatta was asked if he could, while sitting in his room in Bombay, tell his listeners what the weather was in NYC. He thought they were nuts and told them so. He said that is a special power and can be developed, but has NOTHING to do with breaking free of the illusion and awakening.
I tend to think that as one works the path of awakening, one develops some skills and once awake, still has those skills. Like lifting weights as part of practice, till one can bench press 300 lbs. Then, one day he awakes and while lifting weight is clearly seen to have nothing to do with Enlightenment, nonetheless the it is taken that E will give one great strength.

But, the question why is the ability to trace previous birth attributed to the Buddha?
I have no idea. It is easy to ascribe it to egoic-Maya illusion - once more (deliberately) confusing E with not-E and once again, as all religions and spiritual practices do - shift one's attention from The Truth, to a "nice story" that will keep one sleeping soundly and feeling good.
The Buddha is quoted as saying "there is suffering, but no one suffers."
How would that square with seeing previous birth?
Sure, images can come and go and stories proliferate.
That's the point. All there IS is the story.

The REAL question is who or what rebirths?

Seems like the right thing to do at the time

When asked "why did you do something," how often have you responded this way. At the time, the path whatever it was, "good or bad" felt "right" and you took it.
Ego insists that "You" are the author of your actions, you control them , you decide what to do.

But is this really true?

For example, where did the idea to do something come from or the idea that a choice needs to be made in the first place?
And, once that choice is presented, WHY does one choice "feel right" and another not right? (I don't mean right and wrong in terms of morality. I mean right and wrong in terms of approach-avoidance.)

If you 're honest and look closely, you cannot say with 100% certainty where the idea came from or why you ended up making a certain choice. You just do.
Best that can be said is that there is witnessing of the choice.
So, while you are certainly the actor, you are not the author.

And if you aren't, then who or what is?